Collusion As Public Monitoring Becomes Noisy

Collusion as Public Monitoring Becomes Noisy: Experimental Evidence∗ Masaki Aoyagi† Guillaume R. Fréchette‡ Osaka University New York University October 13, 2005 Abstract This paper studies collusion in an infinitely repeated game when the oppo- nent’s past actions are observed only through a noisy public signal. Attention is focusedonathreshold strategy, which switches between cooperation and punishment phases based on the comparison between the realized public signal and a threshold. The paper develops a simple theory on when such a strategy supports the most efficient symmetric (perfect public) equilibrium, and characterizes its payoff as a function of noise in monitoring. The theoretical predictions are then tested in laboratory experiments. It is found that subjects’ payoffs (i) decrease as noise increases, and (ii) are lower than the theoretical maximum for small noise, but exceed it for large noise. It is also estimated that the subjects’ strategies are best described by a simple threshold strategy that looks only at the most recent public signal. Keywords: Repeated games, collusion, cooperation. ∗Special thanks are given to John Kagel, who helped with comments and guidance through- out the development of the paper. We have benefited from discussions with Al Roth and Marco Casari and comments by Pedro Dal Bo, Georg Weizsacker and seminar participants at Université Laval, Harvard University, New York University, Université de Montréal, Carnegie Mellon Univer- sity, RIETI, Rutgers, and the Stockholm School of Economics, as well as participants of the 2003 International Meeting of the Economic Science Association, and the 14th Summer Festival on Game Theory at Stony Brook. Alex Brown provided valuable research assistance. This research was sup- ported in part by the Harvard Business School, by the Center for Experimental Social Science and the C.V. Starr Center. We are responsible for all remaining errors. †ISER, 6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan. ‡Department of Economics, 269 Mercer St. 7th Floor, New York, NY, 10003-6687. 1 1Introduction Imperfection in monitoring is an integral part of many competitive situations in reality. It is a particularly important topic in the field of industrial organization, where the signals are subject to various external shocks. Firms’ ability to cooperate in such environments is of clear interest to researchers as well as regulation author- ities. A key prediction of the theory of repeated games is that when players are patient, a simple strategy can sustain collusion even under imperfect public monitoring, where opponents’ actions are observed only through a noisy public signal. In the repeated prisoners’ dilemma (PD), for example, it is known that any payoff in an important class of symmetric equilibria can be achieved through the use of a grim-trigger strategy, which reverts to the one-shot Nash equilibrium in the event of particular signal realizations. This paper is aimed at testing the theory of imperfect public monitoring in infinitely repeated games: It provides a theoretical characterization of the maximal equilibrium payoff as a function of noise in monitoring, and then uses laboratory experiments to test the theoretical predictions. The main focus of the paper is on the comparative statics of the effect of noise on the players’ payoffs, and on the strategies they use to achieve collusion. While imperfect monitoring has attracted much attention in economic theory, empirical work on the subject has been limited because of some fundamental difficulties. For example, it is not easy to identify the exact public signal the firms use to coordinate their actions: it could be price, shares in a nationwide or regional market, industry output, or the combination of any of these and other indicators. There are also difficulties with identifying the firms’ strategic variables and their payoff structure. Free from these problems, a laboratory experiment in a controlled environment is considered an ideal alternative for the study of the subject. Green and Porter (1984) are the first to provide a theoretical analysis of repeated games with imperfect public monitoring: In their model of quantity-setting oligopoly, the market price serves as a noisy public signal of firms’ output quantities because of demand fluctuations. They put forth an equilibrium based on the trigger strategy as follows: The firms produce at the jointly monopolistic level as long as the realized price is above a certain threshold, but revert to the one-shot Cournot quantity for a fixed number of periods when it falls below the threshold. Because of the random component in demand, periodic price wars occur on the equilibrium path. 2 As in Green and Porter (1984), the present model has a one-dimensional public signal (price) whose distribution is monotonically related to the players’ actions (quantities). While the set of repeated game strategies is large, we suppose that players coordinate their actions based on a simple rule. In particular, we suppose that players use a threshold strategy, which uses thresholds on the public signal as a coordination device: The players agree to take one action profile if the realized public signal is above a certain value but take another action profile if it falls below it. Specifically, we consider a symmetric stage game with two actions for each player: As in the PD, the first action gives rise to an efficient symmetric profile, while the second action gives rise to an inefficient symmetric (one-shot) Nash equilibrium. We characterize the set of equilibrium payoffs when the two players’ repeated game strategies are symmetric, sequentially rational, and public in the sense that today’s actions are determined only by past public signals. In this case, it is known from the bang-bang property (Abreu et al. (1990)) that the highest equilibrium payoff is sustained by a grim-trigger strategy.Basedonthisfact,weidentifysufficient conditions for the optimal grim-trigger strategy to trigger the punishment based on a threshold condition. We then proceed to derive an explicit and testable link between the maximal symmetric equilibrium payoff and the level of noise in monitoring.1 In our experiments, we use a simple two-person two-action PD as the stage game, and assume a simple distribution for the noise component of the public signal. The distribution allows for an explicit derivation of the maximum payoff as a function of noise in monitoring. We have five treatments depending on the level of noise from zero to infinity. In theory, positive cooperation is possible for the three low noise treatments, while the best equilibrium entails no cooperation for the two high noise treatments. We have two major objectives in analyzing data from our experiments. First, we compare the players’ actual payoffs against the theoretical maximum de- rived as above, and examine how they change with the noise in monitoring. Our findings are as follows: 1) The level of cooperation is positive for any noise level. 2) The level of cooperation is lower than the theoretical maximum for small noise 1 This should be contrasted with the qualitative conclusion of Kandori (1992) that the set of (symmetric and asymmetric) perfect public equilibrium payoffs strictly expands as monitoring becomes more accurate. We are unaware of any quantitative characterization of equilibrium payoffs as a function of noise in monitoring. 3 but higher than that for large noise. 3) The level of cooperation decreases as noise increases. 4) For large noise for which theory predicts no cooperation, the level of cooperation is no higher than that in a one-shot game. We also find a distinctive pattern in the evolution of play. In the low noise treatments for which cooperation is possible in theory, we observe that subjects increase the level of cooperation as they accumulate experience. In the high noise treatments, on the other hand, the subjects begin to behave more non-cooperatively as they become more experienced. As stated above, we find that reducing noise in monitoring increases the level of cooperation. It should be emphasized that this is the first experiment to identify such a relationship between the noise in monitoring and the level of cooperation in the standard imperfect public monitoring setting. This result is far from obvious for the following reasons: First, as is well known, experiments on the one-shot PD often generate positive levels of cooperation. Given that their subjects cooperate even with no histories, these experiments may be interpreted as suggesting that past histories would play an insignificant role in their decisions in repeated games. The present experiment rejects this view. Second, our finding is at odds with those of some experiments on imperfect monitoring. These experiments, which model imperfect monitoring in ways significantly different from the present one, find no increase in cooperation when monitoring becomes “more accurate.”2 Third, sustaining cooperation in a repeated game requires the ability to perform non-trivial reasoning even if it only involves a simple strategy. Our finding suggests that the subjects do in fact possess such capabilities. Our second objective is to estimate the subjects’ repeated game strategies. For this, we identify the strategy that best describes the data from a class of strategies that use thresholds. Specifically, we consider a class of strategies which start out in the cooperation phase, switch to the punishment phase when the public signal falls below a certain threshold, and return to the cooperation phase after some number of periods provided that the signal then is above another threshold. It can be seen that this class encompasses those strategies that are most frequently discussed in the literature: Included in this class are the trigger strategies as in Green and Porter 2 As discussed in the next section, these experiments deviate in some important ways from the standard models of imperfect public monitoring.

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